If you are migrating your existing site from Joomla 1.0 to 1.5, you must make sure that any user IDs remain the same on the new site, otherwise, when the nBill tables are copied over, the client records may be associated with the wrong users (thus giving people access to other people's invoices, etc.).
Other than that, it should be quite straightforward, as you say, just install a fresh copy of nBill (or copy the nBill files from your old site), and copy the database tables over. After copying the tables over, the first time you try to use nBill administrator, it will prompt you for the key (because the old key will have been copied over) - using a free trial key is fine while you set things up.
The only other thing to look out for is that Joomla 1.5 uses UTF-8 character encoding rather than iso-8859-2, so if you have any accented characters or special symbols (eg. currency symbols), they might get a bit garbled so you may have to correct them.